Suppose that $A$ is an $n\times n$ square matrix and $S=\set{\vectorlist{x}{p}}$ is a set of eigenvectors with eigenvalues $\scalarlist{\lambda}{p}$ such that $\lambda_i\neq\lambda_j$ whenever $i\neq j$. Then $S$ is a linearly independent set.
There is a simple connection between the eigenvalues of a matrix and whether or not the matrix is nonsingular.
Suppose $A$ is a square matrix. Then $A$ is singular if and only if $\lambda=0$ is an eigenvalue of $A$.
With an equivalence about singular matrices we can update our list of equivalences about nonsingular matrices.
Suppose that $A$ is a square matrix of size $n$. The following are equivalent.
Certain changes to a matrix change its eigenvalues in a predictable way.
Suppose $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Then $\alpha\lambda$ is an eigenvalue of $\alpha A$.
Unfortunately, there are not parallel theorems about the sum or product of arbitrary matrices. But we can prove a similar result for powers of a matrix.
Suppose $A$ is a square matrix, $\lambda$ is an eigenvalue of $A$, and $s\geq 0$ is an integer. Then $\lambda^s$ is an eigenvalue of $A^s$.
While we cannot prove that the sum of two arbitrary matrices behaves in any reasonable way with regard to eigenvalues, we can work with the sum of dissimilar powers of the same matrix. We have already seen two connections between eigenvalues and polynomials, in the proof of Theorem EMHE and the characteristic polynomial (Definition CP). Our next theorem strengthens this connection.
Suppose $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Let $q(x)$ be a polynomial in the variable $x$. Then $q(\lambda)$ is an eigenvalue of the matrix $q(A)$.
Inverses and transposes also behave predictably with regard to their eigenvalues.
Suppose $A$ is a square nonsingular matrix and $\lambda$ is an eigenvalue of $A$. Then $\lambda^{-1}$ is an eigenvalue of the matrix $\inverse{A}$.
The proofs of the theorems above have a similar style to them. They all begin by grabbing an eigenvalue-eigenvector pair and adjusting it in some way to reach the desired conclusion. You should add this to your toolkit as a general approach to proving theorems about eigenvalues.
So far we have been able to reserve the characteristic polynomial for strictly computational purposes. However, sometimes a theorem about eigenvalues can be proved easily by employing the characteristic polynomial (rather than using an eigenvalue-eigenvector pair). The next theorem is an example of this.
Suppose $A$ is a square matrix and $\lambda$ is an eigenvalue of $A$. Then $\lambda$ is an eigenvalue of the matrix $\transpose{A}$.
If a matrix has only real entries, then the computation of the characteristic polynomial (Definition CP) will result in a polynomial with coefficients that are real numbers. Complex numbers could result as roots of this polynomial, but they are roots of quadratic factors with real coefficients, and as such, come in conjugate pairs. The next theorem proves this, and a bit more, without mentioning the characteristic polynomial.
Suppose $A$ is a square matrix with real entries and $\vect{x}$ is an eigenvector of $A$ for the eigenvalue $\lambda$. Then $\conjugate{\vect{x}}$ is an eigenvector of $A$ for the eigenvalue $\conjugate{\lambda}$.
This phenomenon is amply illustrated in Example CEMS6, where the four complex eigenvalues come in two pairs, and the two basis vectors of the eigenspaces are complex conjugates of each other. Theorem ERMCP can be a time-saver for computing eigenvalues and eigenvectors of real matrices with complex eigenvalues, since the conjugate eigenvalue and eigenspace can be inferred from the theorem rather than computed.